Linking Score Approximation to Distribution Accuracy in Diffusion Models

This paper addresses a fundamental theoretical gap in score-based diffusion models, which have achieved significant success in generative AI. While it's known that neural networks can approximate score functions, the direct relationship between this approximation accuracy and the quality of the generated probability distributions has been unclear. The research rigorously quantifies this connection, proving that if a neural network accurately approximates the true score function, the probability distribution generated by the corresponding reverse diffusion process will be close to the target data distribution. This closeness is measured in Kullback-Leibler (KL) divergence, with an explicit upper bound derived for the approximation error. The analysis combines established theorems like Hornik's universal approximation theorem and Girsanov's theorem, providing a clear theoretical foundation. This work complements existing research by offering an approximation-theoretic analysis based on classical neural network theory, providing a crucial guarantee that links the ability of neural networks to learn score functions to their capacity to generate high-fidelity data distributions.